Question

I have to enter into a program called goformative the interval notation for -1 < p < = -6. Another tutor said the answer is empty set or null set. I have tried entering this and it says it is wrong. Is there something missing?

Answer 1

The inequalities to solve are,

$\begin{gathered} 7p+5\leq-37 \\ \text{and} \\ -10p<10 \end{gathered}$

Solving the first inequality:

$\begin{gathered} 7p+5\leq-37 \\ 7p\leq-37-5 \\ 7p\leq-42 \\ p\leq-\frac{42}{7} \\ p\leq-6 \end{gathered}$

Solving the second inequality:

$\begin{gathered} -10p<10 \\ \frac{-10p}{-10}>\frac{10}{-10} \\ p>-1 \end{gathered}$

So, the solution is

$\begin{gathered} p\leq-6 \\ \text{and} \\ p>-1 \end{gathered}$

The graph of both solutions are shown below:

$p\leq-6$

Then,

$p>-1$

Now, recall that in interval notation, "(" and ")" indicates that an interval is not included, and "[" and "]" indicates that an interval is included.

We can write both solutions in interval notation as,

$\begin{gathered} p\leq-6 \\ \text{and} \\ p>-1 \\ --------------- \\ \text{Interval Notation:} \\ (-\infty,-6\rbrack \\ \text{and} \\ (-1,\infty) \\ --------------- \\ (-\infty,-6\rbrack U(-1,\infty) \end{gathered}$